Henceforth, we will vectorize all the image variables \(u \) , \(u_0\)  and the auxiliary variable \(y\)  including \(l\) , \(b\)  and \(s\)  as \(u\in \mathbb {R}^{MN}\) , \(u_0\in \mathbb {R}^{M_0N_0}\)  and \(y\)  including \({l}\) , \({b}\)  and \({s}\)  in the corresponding spaces and for the corresponding models. Besides, we will still use the linear operators \(A\) , \(\nabla _i\) , \(i=1,2\)  as their corresponding matrix versions after vectorizing all the variables. Now, we will touch some necessary tools from convex and variational analysis [1]}, [2]}, [3]}.
The graph of a multivalued mapping \(F : R^n \Rightarrow R^m\)  is defined by
\(\operatorname{gph}F := \lbrace (x,y) \in R^n \times R^m : y \in F(x)\rbrace ,\) 
